Backwards Analysis Gerth Stølting Brodal Pre-talent track activity in Algorithms, Department of Computer Science, Aarhus University, February 19, 2019
Pareto optimal / non-dominated points / skyline No points Pareto optimal Dominated points Vilfredo Pareto (1848 – 1923) Skyline
Exercis ise 1 (skyline construction) Given n points ( x 1 , y 1 ), ..., ( x n , y n ) in R 2 Give an algorithm for computing the points on the skyline ? What is the running time of your algorithm ? skyline Skyline
Problem – expected skyline siz ize ? ? Consider n points ( x 1 , y 1 ), ..., ( x n , y n ) in R 2 Each x i and y i is selected independently and uniformly at random from [0, 1] 1 What is the expected skyline size ? 0 0 1 Skyline size
Exercis ise 2 (dependent points) Describe an algorithm for generating n distinct points ( x 1 , y 1 ), ..., ( x n , y n ) in R 2 Each x i and y i is selected uniformly at 1 random from [0, 1] The points are not independent 0 0 1 Skyline size
Generating random points Assume the points are generated by the following algorithm 1) Generate n random x -values x 1 , ..., x n 2) sort the x -values in decreasing order 1 3) for decreasing x i generate random y i ( x i , y i ) ( x i , y i ) is a skyline point ⟺ y i = max( y 1 , ..., y i ) Prob[( x i , y i ) skyline point] = 1 i since y 1 , ..., y i independt and the same distribution, ( x 1 , y 1 ) all permutations are equally likely, i.e. probability 0 y i to be largest among i values is 1/ i 0 1 Skyline size
Expected skyline siz ize Stochastic variable: Xi = ቊ 1 if point i on skyline 0 otherwise E[|skyline|] 1 = E[ X 1 + ∙∙∙ + X n ] = E[ X 1 ] + ∙∙∙ + E[ X n ] ( x i , y i ) = Σ i Prob[( x i , y i ) skyline point] = 1 1 + 1 2 + ∙∙∙ + 1 n ( x 1 , y 1 ) 1 = σ i =1.. n (harmonic number H n ) 0 i 0 1 ≤ ln n + 1 Skyline size
n Σ n -th Harmonic number H n = 1/1 + 1/2 + 1/3 +∙∙∙+ 1/ n = 1/ i i = 1 ∫ n n H n – 1 H n – 1/n 1/ x dx = [ ln x ] = ln n – ln 1 = ln n 1 1 ln n + 1/ n H n ln n + 1 1/ n 1/1 1/2 1/3 1/4 1/ n 1 2 3 4 5 n Harmonic numbers
Exercis ise 3 (divide-and-conquer skyline) Consider the following algorithm Find the topmost point p in O( n ) time recurse on points to the right of p 1 p Show that the expected running time is O( n ) 0 0 1 Skyline
QuickSort – a a randomized sorting alg lgorithm Quicksort( x 1 , ..., x n ) Pick a random pivot element x i Partition remaining elements: S smaller than x i , and L larger than x i Recursively sort S and L 7 4 15 12 8 11 3 5 1 14 6 7 4 3 5 1 6 8 15 12 11 14 3 1 4 7 5 6 11 15 12 14 1 3 5 6 7 12 14 15 3 5 6 12 15 6 QuickSort
QuickSort – analy lysis I Alternative Quicksort Consider a random permutation π of the input, such that x π (1) is the first pivot, then x π (2) , x π (3) , .... Observations Changes the order unsorted sublists are partitioned, but the pivots are still selected uniformly among the elements x i and x j are compared if and only x i or x j is selected as a pivot before any element in the sorted list between x i and x j output x j x i QuickSort
QuickSort – analy lysis II II E[comparisons by quicksort] = E[ Σ i < j cost of comparing x i and x j ] = Σ i < j E[cost of comparing x i and x j ] 2 = Σ i < j | r j − r ( i )| + 1 where r ( i ) = position of x i in output 2 = Σ 1≤ p < q ≤ n q − p + 1 1 ≤ 2 n ∙Σ 2≤ d ≤ n r ( i ) output r ( j ) d x j x i ≤ 2 n ∙(( ln n + 1) – 1) q p = 2 n ∙ln n d QuickSort
Exercis ise 4 (random search trees) Construct a unbalanced binary search tree for n numbers x 1 <∙∙∙< x n by inserting the numbers in random order What is the probability that x j is an ancestor of x i ? 15 What is the expected depth of a node x i ? 8 17 13 3 5 10 Insert: 15, 8, 17, 13, 3, 5, 10 Random search trees
Convex hull Convex hull = smallest polygon enclosing all points Convex hull
Exercis ise 5: Convex hull Give an O( n ∙log n ) worst-case algorithm finding the Convex Hull Convex hull
Convex hull – randomized in incremental 1) Let p 1 , ..., p n be a random permutation of the points 2) Compute convex hull of { p 1 , p 2 , p 3 } 3) c = ( p 1 + p 2 + p 3 ) / 3 4) for i = 4 to n insert p i and construct H i from H i -1 : if p i inside H i -1 skip, otherwise insert p i in H i -1 and delete chain inside p 3 p i c c p 2 p 1 H 3 H i -1 Convex hull
Convex hull – in inserting p i p i e c H i -1 Convex hull
Convex hull – in inserting p i p i e c How to find e for p i ? store set of points with e H i -1 and reference to e from p i Convex hull
Convex hull - analy lysis Each point inserted / deleted / inside at most once in convex hull p E[# point-edge updates] u = E[ Σ 4≤ i ≤ n Σ p p updated on insertion i ] e = Σ 4≤ i ≤ n Σ p E[ p updated on insertion i ] v 2 ≤ Σ 4≤ i ≤ n Σ p i − 3 ≤ 2 n ∙( ln n + 1) c since p only updated on insertion i if p i is u or v H i Total expected time O( n ∙log n ) Convex hull
Bin inary ry search - but forgot to sort the array... (a debugg gging case) How many cells can ever be 17 reached by a binary search ? useful 31 43 useful 3 13 47 29 useful not reachable 2 41 11 7 23 19 5 37 find(41) 2 3 41 31 11 13 7 17 23 47 19 43 5 29 37 Binary searching unsorted array
Reachable nodes – analy lysis Pr[ v i useful ] = | L i | / Σ j | L j | E[ # useful nodes at level ] = Σ i ( | L i | / Σ j | L j | ) = 1 E[ # useful nodes in tree ] = height - 1 E[ # reachable nodes in tree ] ≤ height 2 = O(log 2 n ) useful 17 useful 31 43 ]- ∞,17[ ∅ ]17,43[ ]43,∞[ # useful nodes ? v 1 v 2 v 3 v 4 L 2 = ∅ L 1 = { 2,3,4,5,11 } L 3 = { 19,23,29,37,41 } L 4 = { 47 } Binary searching unsorted array
Clo losest pair ir p q Given n points, find pair ( p , q ) with minimum distance Algorithm (idea) permute points randomly p 1 , p 2 , ..., p n for i = 2.. n compute Δ i = distance between closest pair for p 1 , ..., p i Observation Pr[ Δ i < Δ i -1 ] ≤ 2 since minimum distance can only decrease i if p i is defining Δ i Closest pair
Clo losest pair ir – grid cell lls Construct grid cells with side-length Δ i -1 Point ( x , y ) in cell ( x /Δ i −1 , y /Δ i −1 ) ≤ 4 points in cell if all pairwise distances ≥ Δ i -1 p i Neighbors of p i within distance Δ i -1 are in ≤ 9 cells Store non-empty cells in a hash tabel (using randomization) Δ i -1 Rebuild grid whenever Δ i decreases Analysis E[rebuild cost] = E[ Σ 3≤ i ≤ n rebuild cost inserting p i ] 2 = Σ 3≤ i ≤ n E[rebuild cost inserting p i ] ≤ Σ 3≤ i ≤ n i i ≤ 2 n Total expected time O( n ) Closest pair
7 0.9 Handin (Treaps) 13 3 0.5 0.1 A treap is a binary search tree with a random priority 21 11 assigned to each element when inserted (in the example 0.4 0.3 elements are white and priorities yellow). 8 A left-to-right inorder traversal gives the elements 0.1 in sorted order, whereas the priorities satisfy heap order, i.e. priorities increase along a leaf-to-root path. a) Argue that an arbitrary element in a treap of size n has expected depth O(log n ) b) Describe how to insert an element into a treap and give running time Treaps
References Raimund Seidel, Backwards Analysis of Randomized Geometric Algorithms , in Pach J. (eds) New Trends in Discrete and Computational Geometry. Algorithms and Combinatorics, vol 10, 37-67. Springer, Berlin, Heidelberg 1993. DOI: 10.1007/978-3-642-58043-7_3 (public available at CiteSeerX). [Chapters 4-5] Sariel Har-Peled, Backwards analysis , lecture notes, 2014. sarielhp.org. [Chapters 39.1-39.3] Raimund Seidel and Cecilia R. Aragon, Randomized search trees , Algorithmica, 16(4 – 5), 464 – 497, 1996. DOI 10.1007/BF01940876. David R. Karger, Philip N. Klein, Robert E. Tarjan, A randomized linear-time algorithm to find minimum spanning trees , Journal of the ACM, 42(2), 321-328, 1995. DOI 10.1145/201019.201022. Timothy M. Chan, Backwards analysis of the Karger-Klein-Tarjan algorithm for minimum spanning trees , Information Processing Letters, 67(6), 303-304, 1998. DOI 10.1016/S0020-0190(98)00129-X. References
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